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Advanced Quantum Field Theory

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Advanced Quantum Field Theory Advanced Quantum Field Theory Roberto Casalbuoni Dipartimento di Fisica Universit`a di Firenze Lezioni date all’Universita’ di Firenze nell’a.a. 2004/2005. Contents Index...................................... 1 1 Notations and Conventions 4 1.1Units.................................... 4 1.2RelativityandTensors.......................... 6 1.3TheNoether’stheoremforrelativisticfields.............. 6 1.4FieldQuantization............................ 9 1.4.1 TheRealScalarField....................... 10 1.4.2 TheChargedScalarField.................... 12 1.4.3 TheDiracField.......................... 14 1.4.4 TheElectromagneticField.................... 17 1.5PerturbationTheory........................... 27 1.5.1 TheScatteringMatrix...................... 27 1.5.2 Wick’stheorem.......................... 29 1.5.3 Feynmandiagramsinmomentumspace............. 30 1.5.4 Thecross-section......................... 33 2 One-loop renormalization 36 2.1DivergencesoftheFeynmanintegrals.................. 36 2.2Higherordercorrections......................... 38 2.3Theanalysisbycounterterms...................... 45 2.4DimensionalregularizationoftheFeynmanintegrals.......... 52 2.5Integrationinarbitrarydimensions................... 54 2.6OneloopregularizationofQED..................... 56 2.7Onelooprenormalization......................... 63 3 Vacuum expectation values and the S matrix 73 3.1In-andOut-states............................ 73 3.2 The S matrix............................... 79 3.3Thereductionformalism......................... 81 4 Path integral formulation of quantum mechanics 85 4.1Feynman’sformulationofquantummechanics............. 85 4.2Pathintegralinconfigurationspace................... 88 1 4.3Thephysicalinterpretationofthepathintegral............ 91 4.4Thefreeparticle............................. 97 4.5Thecaseofaquadraticaction...................... 99 4.6Functionalformalism...........................104 4.7Generalpropertiesofthepathintegral.................107 4.8ThegeneratingfunctionaloftheGreen’sfunctions...........113 4.9 The Green’s functions for the harmonic oscillator ...........116 5 The path integral in field theory 123 5.1Thepathintegralforafreescalarfield.................123 5.2 The generating functional of the connected Green’s functions . 127 5.3 The perturbative expansion for the theory λϕ4 .............129 5.4TheFeynman’srulesinmomentumspace................135 5.5 Power counting in λϕ4 ..........................137 5.6 Regularization in λϕ4 ...........................142 5.7 Renormalization in the theory λϕ4 ...................145 6 The renormalization group 152 6.1Therenormalizationgroupequations..................152 6.2Renormalizationconditions........................156 6.3ApplicationtoQED...........................160 6.4Propertiesoftherenormalizationgroupequations...........161 6.5 The coefficients of the renormalization group equations and the renor- malizationconditions...........................167 7 The path integral for Fermi fields 169 7.1 Fermionic oscillators and Grassmann algebras .............169 7.2IntegrationoverGrassmannvariables..................174 7.3Thepathintegralforthefermionictheories...............178 8 The quantization of the gauge fields 180 8.1QEDasagaugetheory..........................180 8.2Non-abeliangaugetheories........................182 8.3Pathintegralquantizationofthegaugetheories............186 8.4PathintegralquantizationofQED...................196 8.5 Path integral quantization of the non-abelian gauge theories . 200 8.6 The β-functioninnon-abeliangaugetheories..............204 9 Spontaneous symmetry breaking 213 9.1 The linear σ-model............................213 9.2Spontaneoussymmetrybreaking.....................219 9.3TheGoldstonetheorem..........................223 9.4TheHiggsmechanism..........................225 9.5 Quantization of a spontaneously broken gauge theory in the Rξ-gauge 230 2 9.6 ξ-cancellationinperturbationtheory..................235 10 The Standard model of the electroweak interactions 239 10.1TheStandardModeloftheelectroweakinteractions..........239 10.2TheHiggssectorintheStandardModel................244 10.3 The electroweak interactions of quarks and the Kobayashi-Maskawa- Cabibbomatrix..............................249 10.4TheparametersoftheSM........................258 10.5Wardidentitiesandanomalies......................259 A 265 A.1Propertiesoftherealantisymmetricmatrices.............265 3 Chapter 1 Notations and Conventions 1.1 Units In quantum relativistic theories the two fundamental constants c e /h, the light velocity and the Planck constant respectively, appear everywhere. Therefore it is convenient to choose a unit system where their numerical value is given by c = /h =1 (1.1) For the electromagnetism we will use the Heaviside-Lorentz system, where we take also 0 =1 (1.2) 2 From the relation 0µ0 =1/c it follows µ0 =1 (1.3) In these units the Coulomb force is given by e e 1 |F | 1 2 = 2 (1.4) 4π |x1 − x2| and the Maxwell equations appear without any visible constant. For instance the gauss law is ∇· E = ρ (1.5) dove ρ is the charge density. The dimensionless fine structure constant e2 α = / (1.6) 4π0hc is given by e2 α = (1.7) 4π 4 Any physical quantity can be expressed equivalently by using as fundamental unit energy, mass, lenght or time in an equivalent fashion. In fact from our choice the following equivalence relations follow ct ≈ =⇒ time ≈ lenght E ≈ mc2 =⇒ energy ≈ mass E ≈ pv =⇒ energy ≈ momentum Et ≈ /h =⇒ energy ≈ (time)−1 ≈ (lenght)−1 (1.8) In practice, it is enough to notice that the product ch/ has dimensions [E· ]. Therefore ch/ =3· 108 mt · sec−1 · 1.05 · 10−34 J · sec = 3.15 · 10−26 J · mt (1.9) Recalling that 1eV=e · 1=1.602 · 10−19 J (1.10) it follows 3.15 · 10−26 ch/ = MeV · mt = 197 MeV · fermi (1.11) 1.6 · 10−13 From which 1MeV−1 = 197 fermi (1.12) Using this relation we can easily convert a quantity given in MeV (the typical unit used in elementary particle physics) in fermi. For instance, using the fact that also the elementary particle masses are usually given in MeV , the wave lenght of an electron is given by · e 1 ≈ 1 ≈ 200 MeV fermi ≈ λCompton = 400 fermi (1.13) me 0.5 MeV 0.5 MeV Therefore the approximate relation to keep in mind is 1 = 200 MeV · fermi. Fur- thermore, using c =3· 1023 fermi · sec−1 (1.14) we get 1fermi=3.3 · 10−24 sec (1.15) and 1MeV−1 =6.58 · 10−22 sec (1.16) Also, using 1 barn = 10−24 cm2 (1.17) it follows from (1.12) 1GeV−2 =0.389 mbarn (1.18) 5 1.2 Relativity and Tensors Our conventions are as follows: the metric tensor gµν is diagonal with eigenvalues (+1, −1, −1, −1). The position and momentum four-vectors are given by xµ =(t, x),pµ =(E,p),µ=0, 1, 2, 3 (1.19) where x and p are the three-dimensional position and momentum. The scalar prod- uct between two four-vectors is given by µ ν µ µ 0 0 a · b = a b gµν = aµb = a bµ = a b − a ·b (1.20) where the indices have been lowered by ν aµ = gµν a (1.21) µν and can be raised by using the inverse metric tensor g ≡ gµν . The four-gradient is defined as ∂ ∂ ∂ ∂µ = = , = ∂t, ∇ (1.22) ∂xµ ∂t ∂x The four-momentum operator in position space is µ ∂ p → i =(i∂t, −i∇ ) (1.23) ∂xµ The following relations will be useful 2 µ →− ∂ ∂ − p = pµp µ = (1.24) ∂x ∂xµ x · p = Et − p · x (1.25) 1.3 The Noether’s theorem for relativistic fields We will now review the Noether’s theorem. This allows to relate symmetries of the action with conserved quantities. More precisely, given a transformation involving both the fields and the coordinates, if it happens that the action is invariant under this transformation, then a conservation law follows. When the transformations are limited to the fields one speaks about internal transformations.Whenboth types of transformations are involved the total variation of a local quantity F (x) (that is a function of the space-time point) is given by ∆F (x)=F˜(x) − F (x)=F˜(x + δx) − F (x) ∼ µ ∂F(x) = F˜(x) − F (x)+δx (1.26) ∂xµ 6 The total variation ∆ keeps into account both the variation of the reference frame and the form variation of F . It is then convenient to define a local variation δF, depending only on the form variation δF(x)=F˜(x) − F (x) (1.27) Then we get ∂F(x) ∆F (x)=δF(x)+δxµ (1.28) ∂xµ Let us now start form a generic four-dimensional action S = d4x L(φi,x),i=1,...,N (1.29) V µ and let us consider a generic variation of the fields and of the coordinates, x = xµ + δxµ ∂φ ∆φi(x)=φ˜i(x) − φi(x) ≈ δφi(x)+δxµ (1.30) ∂xµ If the action is invariant under the transformation, then S˜V = SV (1.31) This gives rise to the conservation equation L L L µ ∂ i − ν ∂ i ∂µ δx + i ∆φ δx i φ,ν = 0 (1.32) ∂φ,µ ∂φ,µ This is the general result expressing the local conservation of the quantity in paren- thesis. According to the choice one does for the variations δxµ and ∆φi,andof the corresponding symmetries of the action, one gets different kind of conserved quantities. Let us start with an action invariant under space and time translations. In the case we take δxµ = aµ with aµ independent on x e∆φi = 0. From the general result in eq. (1.32) we get the following local conservation law L µ ∂ i −L µ µ Tν = i φ,ν gν ,∂µTν = 0 (1.33) ∂φ,µ Tµν is called the energy-momentum tensor of the system. From its local conservation we get four constant of motion 3 0 Pµ = d xTµ (1.34) Pµ is the four-momentum of the system. In the case of internal symmetries we take δxµ = 0. The conserved current will be µ ∂L i ∂L i µ J = i ∆φ = i δφ ,∂µJ = 0 (1.35) ∂φ,µ ∂φ,µ 7 with an associated constant of motion given by Q = d3xJ 0 (1.36) In general, if the system has more that one internal
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